Ratios
What is a Ratio?A ratio is a comparison of two numbers or measures that have the same unit. The numbers in a ratio are sometimes called terms.
How Can Ratios Be Expressed?
Example: 4 to 3
This ratio can be expressed as 4 to 3, 4:3, or 4/3
What Is an Equivalent Ratio?
Equivalent ratios are ratios that can be represented by equivalent fractions.
How Do We Express Equal Ratios?
Two ratios are equal if they name the same value.
To write equal ratios multiply both terms by the same number or divided both terms by the same number.
Rates
What is Rate?A rate is a ratio between two quantities of different units, likes miles and hours, or dollars and gallons.
Example:
60 miles per hour and $1.20 per gallon are rates.
What is Unit Rate?
When the second number of a rate is 1, then the rate is a unit rate.
Example:
Speed is a rate. The speed of 60 miles per hour compares 0 miles to 1 hour.
Since 1 is the second number in the rate 60 miles per hour = a unit rate.
What is a Unit Price?
Unit price is the cost per unit.
Unit price can be used to help you find the better buy for prices of things.
Example:
A 12oz bottle of ketchup costs $1.29
A 16 oz bottle of ketchup costs $1.85
Which is the better buy?
12oz for $1.29
$1.29/120z = 10.75 cents per oz.
16oz for $1.85
$1.85/16 0z = 11.75 cents per ounce.
The ketchup in the 12ox bottle is the better buy.
Proportions
What is a Proportion:A proportion is an equation stating that two ratios are equivalent.
Example: 6/9 = 24/36
How do we solve proportions using equivalent fractions?
To solve a proportion using equivalent fractions, ask yourself what numbers each term are being multiplied or divided by.
Example:
12/24 = n/12
The second ratio is being divided by 2, so divide 12 by 2.
12/24 = 6/12
n/15 = 6/30
The first ratio is being multiplied by 2, so think what number can be multiplied by 2 to get 6.
2/15 = 6/30
Solving Proportions Using Cross Multiplication
In a proportion the product of the means (the second and third terms of the proportion) equals the produce of the extremes (the first and last terms of a proportion)
a/b = c/d
b x c = a x d
Means Extremes
When you multiply the means and extremes they should be equal. This process is called cross multiplication. Cross multiplication can be used to find a missing term in proportion.
Example: Find The Missing Term
n/9 = 12/4
Step 1: Cross Multiply
4n = 108
Step 2: Divided Each Side By n to solve.
4n/4 = 108/4
n = 27
Step 3: Fill in Term in Proportion
27/9 = 12/4
Step 4: Cross Multiply to Check
27 x 4 = 108
9 x 12 = 108
Percents
What is a percent?
Percent means “out of one hundred.”
The symbol for percent is %.
Percent can be thought of as the ratio of a number to 100, or the numerator of a fraction in which the denominator is 100.
Percents can be represented as fractions and decimals.
Example:
60% = 60/100 or 3/5 or 0.6
40 % = 40/100 or 2/5 or 0.4
How do we solve percent problems?
Decimal Method:
Find 50% of 80
Step 1: Change the percent to an equivalent decimal.
50% = 0.5
Step 2: Multiply:
0.5 x 80 = 40
50% of 80 is 40
Fraction Method:
Find 50% of 80
Step 1: Change the percent of a fraction in lowest terms.
50% = 50/100 or 1/2
Step 2: Multiply:
1/2 x 80 = 40
Part/Whole Method (Rate & Base):
Rate: Percentage
Part: Part of Whole
Base: Total
So use the proportion: part/base = percentage/100
The percent is always over 100, because percent means “out of 100.”
Finding The Percent:
What percent of 30 is 6?
30 is the base; 6 is the part; the percent is what is missing.
Step 1: Set up proportion: 6/30 = n/100
Step 2: Cross Multiply
600 = 30n
Step 3: Divide to Solve Proportion
n = 20
6 is 20% of 30.
Finding The Whole:
60 is 48% of what number?
60 is the part; 48 is the percent; the base is what is missing.
Step 1: Set up proportion: 60/n = 48/100
Step 2: Cross Multiply
6,000 = 48n
Step 3: Divide to Solve Proportion
n = 125
60 is 48% of 125
How do we calculate discounts and bonuses?
To calculate a discount or bonus, you need to find how much of a percent of the whole the discount is, and then add or subtract that amount from the original number.
Example:
The regular price of a pair of shorts is $40.00. During a sale, the store gives a discount of 25%. How much will the price of the shorts be?
Step 1: Find 25 % 94 $40
40 x .25 or 40 x 25/100 = 10.00
Step 2: Subtract the discount from the original prince.
40.00-10.00 = 30.00
The shorts will cost you $30
Example:
Ms. Smith earns $50 for teaching Saturday School. This week Principal Neering is giving Ms, Smith a 25% bonus. How much money will she make this week?
Step 1: Find 25% of 50
50 x .25 or 25/100 = 25.00
Step 2: Add the bonus to the original amount.
50.00 + 2500 = 75.00
Ms. Smith will make $75 this week.
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